Runge&#x27;s Method and Modular Curves - Mathematics > Number Theory

Abstract: We bound the j-invariant of S-integral points on arbitrary modular curvesover arbitrary fields, in terms of the congruence group defining the curve,assuming a certain Runge condition is satisfied by our objects. We then applyour bounds to prove that for sufficiently large prime p, the points of $X 0^+p^rQ$ with r>1 are either cusps or CM points. This can be interpreted asthe non-existence of quadratic elliptic Q-curves with higher prime-powerdegree.